Chirality, confinement and dimensionality govern re-entrant transitions in active matter

TL;DR

This work addresses how chirality, confinement, and dimensionality shape the non-equilibrium steady states of active particles. By solving the Fokker-Planck equation with a Laplace-transform approach for trapped 2D chiral ABPs and 3D torque-driven ABPs, the authors derive exact expressions for displacement moments and the steady-state excess kurtosis, validated by simulations. They identify three regimes—bimodal off-center, Gaussian-like, and weakly heavy-tailed—whose presence and boundaries depend on trap strength, activity, and chirality, with dimensionality playing a decisive role: 2D chirality can suppress activity, while 3D torque preserves it and induces anisotropy, both describable by simple active length-scale arguments. The results yield concrete experimental signatures, such as kurtosis crossovers and anisotropic steady states, and establish confinement as a powerful tool to probe and control chiral and torque-driven active matter, with potential realizations in L-shaped colloids and chiral rotors.

Abstract

The non-equilibrium dynamics of individual chiral active particles underpin the complex behavior of chiral active matter. Here we present an exact analytical framework, supported by simulations, to characterize the steady states of two-dimensional chiral active Brownian particles and three-dimensional torque-driven counterparts in a harmonic trap. Using a Laplace-transform approach of the Fokker-Planck equation, we derive closed-form expressions for displacement moments and excess kurtosis, providing a precise probe of non-Gaussian statistics. Our analysis reveals three distinct regimes: bimodal active states with off-center peaks, Gaussian-like passive states, and weakly heavy-tailed distributions unique to two dimensions. We show that dimensionality plays a decisive role: in two dimensions, increasing chirality suppresses activity and restores passive behavior, while in three dimensions torque preserves activity along the torque axis, producing anisotropic steady states. These behaviors are captured by simple active length-scale arguments that map the boundaries between passive and active phases. Our results offer concrete experimental signatures - including kurtosis crossovers, off-center peaks, and torque-induced anisotropy - that establish confinement as a powerful tool to probe and control chiral and torque-driven active matter.

Figures (3)

• Figure 1: Schematic representation of torque effects in two- and three-dimensional active particles, illustrating their typical steady-state trajectories under harmonic confinement.
• Figure 2: Steady state of a 2D chiral active Brownian particle (cABP) in a harmonic trap at ${\rm{Pe}} = 10^2$. (a, b) Steady-state excess kurtosis $\tilde{\mathcal{K}}_{\rm st}$ as a function of (a) trap strength $\beta$ for $\Omega = 0.1$ ($\circ$), $5$ ($\square$), and $10^4$ ($\triangleright$); and (b) chirality $\Omega$ for $\beta = 0.15$ ($\circ$), $10$ ($\square$), and $10^4$ ($\triangleright$). Symbols: simulation; solid lines: analytical results. (c) Phase diagram in the $(\Omega, \beta)$ plane, with the color map indicating $\tilde{\mathcal{K}}_{\rm st}$. Colored regions denote dynamical phases with bimodal (red), heavy-tailed (green), and Gaussian-like (blue) distributions. Regions with $\tilde{\mathcal{K}}_{\rm st} > 0$ reflect weakly heavy-tailed active behavior. The black contour ($\tilde{\mathcal{K}}_{\rm st} = 0$) marks the Gaussian limit. The dashed lines represent Eq. \ref{['eq_beta']} and its two asymptotic limits: $\beta = {\rm{Pe}}^2/2$ and $\beta = 2\Omega^2/{\rm{Pe}}^2$. Numerical labels denote points where radial distributions are computed in (d–f). (d–f) Radial probability distributions $\rm p(\tilde{\rm{ r}}')$ plotted versus $\tilde{\rm r}' = |{\tilde{\textbf{r}}}|/{\rm{Pe}}$: (d) Varying $\beta$ at fixed $\Omega = 5$ [points (i)–(iv) in (c)]; (e) Varying $\Omega$ at fixed $\beta = 10$ [points (I)–(V) and (iii)]; (f) Varying $\Omega$ at $\beta = 0.15$ [points (1)–(5)]. Symbols: simulation; solid lines: Gaussian reference. Off-center, non-Gaussian peaks at (I), (II), and (III)/(iii) correspond to $\tilde{\rm{ r}}' = \tilde{\rm{ r}}_{\rm ac}/{\rm{Pe}}$.
• Figure 3: Steady state of a 3D active Brownian particle (ABP) under torque in a harmonic trap at ${\rm{Pe}} = 10^2$. (a, b) Steady-state excess kurtosis as a function of $\beta$ and $\Omega$. Symbols: simulation; solid lines: analytical predictions. (c) Phase diagram in the $\Omega$–$\beta$ plane, with color indicating the steady-state excess kurtosis. The lines represent Eq. \ref{['eq_beta']} along with its two limiting cases: $\beta = {\rm{Pe}}^2/2$ and $\beta = 2\Omega^2/{\rm{Pe}}^2$. Also shown are the boundary values at $\beta = 1$ and $\beta = {\rm{Pe}}^2$. (d, e) Probability distribution function plotted as a function of $\tilde{\rm r}_\perp$ and $\tilde{z}$. In (d), $\Omega$ is varied at fixed $\beta = 20$, corresponding to points (i)–(v) in (c). In (e), $\beta$ is varied at fixed $\Omega = 30$, corresponding to points (1), (2), (iii), (4), and (5) in (c).